Betting on how
many prime factors a number has

Will you take this bet?

You pay one dollar to play.

Your task is, given a number between one and 10 million, which will be generated on your calculator using randInt(1, E7), to guess how many distinct prime factors the number has. If you guess correctly, you get three dollars, thus earning two dollars. If you guess incorrectly, you lose your dollar. (You don't get to see the number until after you bet!)

What do you think? How can you sensibly decide whether to play?

Hint. Try http://www.wolframalpha.com. Generate a few such numbers on your calculator and ask Wolfram Alpha to factor them, and see how many factors you get.

Examples

4940664=2^3*3*41*5021 |
4 distinct factors |

5566115=5*23*29*1669 |
4 distinct factors |

9564142 = 2*7*29*23557 |
4 distinct factors |

3858649=193*19993 |
2 |

9062410=2*5*7*37*3499 |
5 |

873619=873619 |
1 (it is prime!) |

1472451=3*467*1051 |
3 |

2618824=2^3*13^3*149 |
3 |

6807953=29*181*1297 |
3 |

1702297=491*3467 |
2 |

925217=925217 |
1 (another prime!) |

2068938=2*3^2*114941 |
3 |

8898764=2^2*7*19*43*289 |
5 |

8434026=2*3^2*468557 |
3 |

4999483=59*84737 |
2 |

Lesson Index